What Quantum Particles Can We Use for Magnetic Resonance?

Quantum Spins and Nuclei

Expected Learning Outcomes

At the end of this module, students should be able to…

  1. calculate the number of spin states and the possible \(m_s\) values for a given spin quantum number, \(s\)

  2. determine whether a nuclear spin will be zero, integer, or half-integer

  3. identify and explain the reasons certain isotopes are most useful for NMR

“The everyday fact that one’s body does not collapse spontaneously into a black hole, therefore, depends on the spin-1/2 of the electron.”

— Malcolm Levitt, Spin Dynamics, pg. 9

quantum spin - a property of quantum particles that has many mathematical similarities to macroscopic spinning objects but also has unique quantum properties not observed in the macroscopic realm

Background Information

Magnetic resonance experiments depend on the interactions between quantum spins and magnetic fields. Here we will introduce quantum spin, explore some of the properties of the spin-1/2 particles that we will primarily be working with, and ultimately determine what particles and nuclei are commonly used in magnetic resonance experiments.

Class-Wide Discussion

quantization - mapping an infinite, continuous set of values into discrete values magnetic moment - also known as magnetic dipole moment or magnetic dipole; the magnetic strength and orientation of a magnet or other object that produces a magnetic field

Discovery of Quantum Spin

In the early twentieth century, physics was undergoing a quantum revolution as more and more of the rules of the quantum world were being discovered. Among these many discoveries was the quantization of physical properties (like the energy levels of an atom). One of the most famous experiments was performed by Otto Stern and Walther Gerlach in 1922. In this experiment Stern and Gerlach aimed to explore the underlying intrinsic magnetic moment that allowed electrically neutral atoms to interact with a magnetic field. A beam of electrically neutral silver atoms were sent through an inhomogeneous magnetic field (where the strength and direction of the field varied over space) and the spatial deflection of the atomic beam was measured by observing where the atoms ended up on a screen after passing through the magnetic field.

Image modified from source (1).

Before viewing the video below, consider the following questions:

Stern-Gerlach Experiment Video Link

Jubobroff, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons. You can find more information on the file information page. FURTHER STUDY: To learn more about the Stern-Gerlach experiment, including the efforts of many scientists to correctly interpret the results as being from the electron spin, see “Right Experiment, Wrong Theory: The Stern-Gerlach Experiment”.

Image modified from source (1).

This experiment provided proof that yet another physical property of the atoms was quantized, and that this property must have an intrinsic magnetic moment that allowed electrically neutral atoms to interact with a magnetic field. Scientists settled upon giving this physical property the name of ‘spin’ because the quantization properties overlapped well with those already known for angular momentum. (We will learn more about angular momentum and other ‘spinning’ aspects of quantum spin very soon!) The Stern-Gerlach apparatus became the primary tool to investigate these quantum spin properties early on - until magnetic resonance techniques were developed.

elementary particles - subatomic particles that make up all known matter and cannot be divided any further into constituent parts bosons - particles with integer spin; like to be buddies (i.e. bosons are happy to all crowd into the same quantum state together) fermions - particles with half-integer spin; like to be frenemies (i.e. two fermions cannot share the same quantum state, but tend to pair up with a fermion with opposite spin)

Spin Quantum Numbers

The quantum spin of a particle, commonly denoted by the spin quantum number \(s\), is one of the few physical characteristics that uniquely identify an elementary particle - along with other information like mass and electric charge. A particle that has different spin - even if all other physical characteristics are the same - can have very different quantum mechanical behavior. Every particle has an associated spin quantum number that is either an integer or half-integer. The set of particles with integer spin are called bosons and the set of particles with half-integer spin are called fermions.

Spin Quantum Number

Commonly denoted by \(s\), which is a positive, dimensionless number. Particles can either have an integer spin (i.e. \(s = 0, 1, 2, ...\)) or a half-integer spin (i.e. \(s = 1/2, 3/2, 5/2, ...\))

Zeeman effect - the effect of splitting of quantum energy levels in the presence of a magnetic field

The Stern-Gerlach apparatus also demonstrates how to separate out different spin states of a quantum system - by putting a quantum spin in an external magnetic field. Interacting with the external magnetic field causes the different quantum spin states to have distinct energy levels, this is known as the Zeeman effect. The energy separation of these spin states is directly proportional to the strength of the applied magnetic field, \(B_0\). These distinct spin states are indexed by the spin magnetic quantum number, \(m_s\), and this quantum number can be negative or positive.

ScientistHP, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons. You can find more information on the file information page.

Spin Magnetic Quantum Number

Commonly denoted by \(m_s\), which is a dimensionless number that goes from \(-s\) to \(s\) in increments of 1.

The number of spin states can be calculated using the spin quantum number of the particle, \(s\):

\[2s + 1\]

For example, if \(s = 1\) then there would be 3 possible spin states, with \(m_s = -1, 0, \textrm{ or } 1\).

Guided Inquiry Questions

  1. How many different allowed spin states does a spin-2 particle have? What are the \(m_s\) values for these states?

  2. How many different allowed spin states does a spin-3/2 particle have? What are the \(m_s\) values for these states?

  3. Quantum spins with \(s = 0\) are sometimes said to occupy a singlet state and spins with \(s = 1\) are sometimes said to occupy a triplet state. What do you think the reasoning is behind those names?

  4. What was the spin quantum number \(s\) of the neutral silver atoms used in the Stern-Gerlach experiment shown above? How did you come to that conclusion?

Potential Sources for Magnetic Resonance

The most relevant quantum particles in our work will be the main components of the atom: electrons, protons, and neutrons. Conveniently, all three have \(s = 1/2\) and are called spin-1/2 particles.

Spin-½ Particles and Their Properties

For spin-1/2 particles (\(s = 1/2\)), there will be two distinct spin states: spin-up (\(m_s = +1/2\) and typically represented by \(\uparrow\)) and spin-down (\(m_s = -1/2\) and typically represented by \(\downarrow\)).

To find the total spin of a combination of multiple spin-1/2 particles, each spin pair contributes zero to the total spin.

As fermions, spin-1/2 particles are excluded from sharing identical quantum states (Pauli exclusion principle.) When fermions are put together (like electrons in atomic orbitals or the protons and neutrons inside the atomic nucleus) they fill up available quantum states by pairing up (spin-up with a spin-down so total contribution to overall spin is zero) before proceeding to the next level (Aufbau principle). The quantum behavior of spin-1/2 particles explains the majority of atomic and molecular structure of matter, so they are he most studied and commonly used quantum spins in experiments.

Guided Inquiry Questions

  1. How is the Aufbau principle used in chemistry for the electron configuration of a carbon atom?

  2. Protons and neutrons inside the atomic nucleus also obey the Aufbau principle. Protons and neutrons are each made up of three quarks which each carry spin-1/2. How does the Aufbau principle explain why the three spin-1/2 quarks add together to give a total spin of 1/2 for protons and neutrons?

isotope - an isotope of a chemical element has the same atomic number (i.e. number of protons) but a different atomic mass (i.e. different number of neutrons); commonly written in the form “carbon-14” or \(^{14}\)C) where the number is the atomic mass of the isotope

Nuclear Spin

For understanding NMR, we are interested in the total nuclear spin of the atomic nuclei, usually denoted by \(I\). Since both protons and neutrons contribute to the nuclear spin, we have to be explicit about what atomic isotope we are observing. If you need a refresher on how to determine the number of proton and neutrons in the nucleus for a given isotope, check out the helpful figure in the margin! NMR only works for isotopes that have non-zero nuclear spin, so to determine whether a given isotope may be a good candidate for NMR, it is helpful to follow some rules to determine the nuclear spin of the isotope.

Rules for finding the nuclear spin of a given isotope

By far the most NMR research is done on spin-1/2 nuclei, but technically NMR can be done on any non-zero spin. In this activity, you get to determine which isotopes may be the best choices for NMR.

Explore the simulation and periodic table below to answer the following questions.

Check out this PhET simulation to explore some of the different isotopes in the first few rows of the periodic table.

Image source (2)

Guided Inquiry Questions

  1. Why do you think it is important to use stable nuclei for NMR experiments?

  2. Which hydrogen isotope do you think is referenced in the periodic table above? Any advantages or drawbacks to choosing to use this particular isotope?

  3. Which carbon isotope do you think is referenced in the periodic table above? Any advantages or drawbacks to choosing this particular isotope?

  4. Which fluorine isotope do you think is referenced in the periodic table above? Any advantages or drawbacks to choosing to use this particular isotope?

Reflection Questions

  1. Why might learning more about quantum spins be important?

  2. Use what you learned about the Aufbau principle and the simplified explanation of how spin-1/2 particles get added together to provide some justification for the rules given above for finding the nuclear spin of an isotope.

  3. For each of the following nuclear isotopes, provide your assessment on whether they may be useful for NMR or not. (Look for non-zero nuclear spin, stability, relative abundance, etc.)

  1. Phosphorus-31 (\(^{31}\)P):

  2. Carbon-15 (\(^{15}\)C):

  3. Helium-3 (\(^3\)He):

  4. Silicon-29 (\(^{29}\)Si):